Mean value problem

In mathematics, the mean value problem was posed by Stephen Smale in 1981. This problem is still open in full generality. The problem asks:


 * For a given complex polynomial $$f$$ of degree $$d \ge 2$$ and a complex number $$z$$, is there a critical point $$c$$ of $$f$$ (i.e. $f'(c) = 0$) such that


 * $$ \left| \frac{f(z) - f(c)}{z - c} \right| \le K|f'(z)| \text{ for }K=1 \text{?} $$

It was proved for $$K=4$$. For a polynomial of degree $$d$$ the constant $$K$$ has to be at least $$\frac{d-1}{d} $$ from the example $$f(z) = z^{d} - d z$$, therefore no bound better than $$K=1$$ can exist.

Partial results
The conjecture is known to hold in special cases; for other cases, the bound on $$K$$ could be improved depending on the degree $$d$$, although no absolute bound $$K<4$$ is known that holds for all $$d$$.

In 1989, Tischler has shown that the conjecture is true for the optimal bound $$K = \frac{d-1}{d} $$ if $$f$$ has only real roots, or if all roots of $$f$$ have the same norm. In 2007, Conte et al. proved that $$K \le 4 \frac{d-1}{d+1}$$, slightly improving on the bound $$K \le 4$$ for fixed $$d$$. In the same year, Crane has shown that $$K < 4-\frac{2.263}{\sqrt{d}}$$ for $$d \ge 8$$.

Considering the reverse inequality, Dubinin and Sugawa have proven that (under the same conditions as above) there exists a critical point $$\zeta$$ such that $$ \left| \frac{f(z) - f(\zeta)}{z - \zeta} \right| \ge \frac{|f'(z)|}{n 4^{n}} $$. The problem of optimizing this lower bound is known as the dual mean value problem.